E19 -- De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt

(On transcendental progressions, that is, those whose general terms cannot be given algebraically)

Summary:

One of Euler’s earlier papers, he starts with Wallis’ "hypergeometric series" 1! + 2! + 3! + 4! + ... (he didn’t have the !-factorial notation yet) to find the value of the function for a general value of x. He defines [x] to be ∫₀¹ (ln(1/t))^x dt, which by substituting t=e^-z, equals ∫₀^∞z^xe^-zdz, or what we know as the Gamma function. He finds [1/2] to be √(p/2) and derives the recursion [x+1] = (x+1)[x]. He finds [p/q] to be a product of beta functions, and derives a differential quotient.

According to the records, it was presented to the St. Petersburg Academy on November 28, 1729.
Euler gave the essential content of this treatise to his friend Goldbach in a letter on January 8, 1730.

Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 5, 1738, pp. 36-57
Opera Omnia: Series 1, Volume 14, pp. 1 - 24
Reprinted in Comment. acad. sc. Petrop. 5, ed. nova, Bononiae 1744, pp. 28-47 [E19a]

Documents Available:

Original publication: E019 (in the Commentarii), Volume 5
English translation (Stacy Langton): E019
German Translation (Alexander Aycock and Arseny Skryagin): E19
The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E19 include:
- Calinger R., “Leonhard Euler: The first St Petersburg years (1727-1741).” Historia Mathematica, 23 (2), pp. 121-166 (May 1996).
- Dutka J., “The early history of the factorial function.” Archive for History of Exact Sciences, 43 (3), pp. 225-249 (1991).
- Ferraro G., “Differentials and differential coefficients in the Eulerian foundations of the calculus.” Historia Mathematica, 31 (1), pp. 34-61 (Feb 2004).
- Ferraro G., “Functions, functional equations, and the laws of continuity in Euler.” Historia Mathematica, 27 (2), pp. 107-132 (May 2000).
- Ferraro G., “Some aspects of Euler's theory of series: Inexplicable functions and the Euler-Maclaurin summation formula.” Historia Mathematica, 25 (3), pp. 290-317 (Aug 1998).
- Ullrich P., “Weierstrass, Karl Lecutre on the History of Elementary-Functions.” Archive for History of Exact Sciences, 40 (2), pp. 143-172 (1989).

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